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3 Trading Strategies Involving Options 239 Table 11.1 ST^K^ K\ < ST < K-2 S T ^ K 2 a bull spread created using calls. long call option &T ^ 1 &T ^ 1 short call option (OT A.2J payoff ST-KI The most aggressive bull spreads are those of type 1. They cost very little to set up and have a small probability of giving a relatively high payoff (= K 2 K]). As we move from type 1 to type 2 and from type 2 to type 3, the spreads become more conservative. Example 11.2 An investor buys for $3 a 3-month European call with a strike price of $3 and sells for $1 a 3-month European call with a strike price of $35. The payoff from this bull spread strategy is $5 if the stock price is above $35, and zero if it is below $3. If the stock price is between $3 and $35, the payoff is the amount by which the stock price exceeds $3. The cost of the strategy is $3 $1 = $2. So the profit is: S T <3 3 < S T < 35 Sr> 35 Profit -2 S T Bull spreads can also be created by buying a European put with a low strike price and selling a European put with a high strike price, as illustrated in Figure Unlike bull spreads created from calls, those created from puts involve a positive up-front cash flow to the investor (ignoring margin requirements) and a payoff that is either negative or zero. Figure 11.3 Profit from bull spread created using put options.

4 24 CHAPTER 11 Bear Spreads An investor who enters into a bull spread is hoping that the stock price will increase. By contrast, an investor who enters into a bear spread is hoping that the stock price will decline. Bear spreads can be created by buying a European put with one strike price and selling a European put with another strike price. The strike price of the option purchased is greater than the strike price of the option sold. (This is in contrast to a bull spread, where the strike price of the option purchased is always less than the strike price of the option sold.) In Figure 11.4, the profit from the spread is shown by the solid line. A bear spread created from puts involves an initial cash outflow because the price of the put sold is less than the price of the put purchased. In essence, the investor has bought a put with a certain strike price and chosen to give up some of the profit potential by selling a put with a lower strike price. In return for the profit given up, the investor gets the price of the option sold. Assume that the strike prices are KI and K 2, with KI < K 2. Table 11.2 shows the payoff that will be realized from a bear spread in different circumstances. If the stock price is greater than K 2, the payoff is zero. If the stock price is less than KI, the payoff is K 2 KI. If the stock price is between KI and K 2, the payoff is K 2 S T. The profit is calculated by subtracting the initial cost from the payoff. Example 11.3 An investor buys for $3 a 3-month European put with a strike price of $35 and sells for $1 a 3-month European put with a strike price of $3. The payoff from this bear spread strategy is zero if the stock price is above $35, and $5 if it is below $3. If the stock price is between $3 and $35, the payoff is 35 ST- The options cost $3 - $1 = $2 up front. So the profit is: Sr< 3 3 < S T < 35 st> 35 Profit S T -2 Figure 11.4 Profit from bear spread created using put options.

5 Trading Strategies Involving Options 241 Table 11.2 ST^K^ K\ < ST < K-2 S T ^ K 2 a bear spread created with put options. long put option short put option -(Ki - S T ) payoff Like bull spreads, bear spreads limit both the upside profit potential and the downside risk. Bear spreads can be created using calls instead of puts. The investor buys a call with a high strike price and sells a call with a low strike price, as illustrated in Figure Bear spreads created with calls involve an initial cash inflow (ignoring margin requirements). Box Spreads A box spread is a combination of a bull call spread with strike prices K] and K 2 and a bear put spread with the same two strike prices. As shown in Table 11.3, the payoff from a box spread is always K 2 K]. The value of a box spread is therefore always the present value of this payoff or (K 2 K])e~ rt. If it has a different value there is an arbitrage opportunity. If the market price of the box spread is too low, it is profitable to buy the box. This involves buying a call with strike price K\, buying a put with strike price K 2, selling a call with strike price K 2, and selling a put with strike price K\. If the market price of the box spread is too high, it is profitable to sell the box. This involves buying a call with strike price K 2, buying a put with strike price K\, selling a call with strike price K\, and selling a put with strike price K 2. It is important to realize that a box-spread arbitrage only works with European options. Many of the options that trade on exchanges are American. As shown in Business Snapshot 11.1, inexperienced traders who treat American options as European are liable to lose money. Figure 11.5 Profit from bear spread created using call options.

6 242 CHAPTER 11 Table 11.3 S T^K, KI < S T < K 2 S T ^ K 2 a box spread. bull call spread Sr-Ki bear put spread payoff Butterfly Spreads A butterfly spread involves positions in options with three different strike prices. It can be created by buying a European call option with a relatively low strike price KI, buying a European call option with a relatively high strike price K 3, and selling two European call options with a strike price K 2 that is halfway between K\ and K 3. Generally, K 2 is close to the current stock price. The pattern of profits from the strategy is shown in Figure A butterfly spread leads to a profit if the stock price stays close to K 2, but gives rise to a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. The strategy requires a small investment initially. The payoff from a butterfly spread is shown in Table Suppose that a certain stock is currently worth $61. Consider an investor who feels that a significant price move in the next 6 months is unlikely. Suppose that the market prices of 6-month European calls are as follows: Strike price ($ ) Call price ($) Figure 11.6 Profit from butterfly spread using call options.

7 Trading Strategies Involving Options 243 Table 11.4 ST^K^ K\ < ST ^ K 2 K 2 < S T < KI ST^ K 3 a butterfly spread. first long call C E^ JT AI C t JT AI 7 C t 7 Oy A] second long call short calls -2(5 r - K 2 ) S T - K 3-2(S T - K 2 ) * These payoffs are calculated using the relationship K^ =.5(K\ + KI). payoff* ST-KI KJ-ST The investor could create a butterfly spread by buying one call with a $55 strike price, buying one call with a $65 strike price, and selling two calls with a $6 strike price. It costs $1 + $5 (2 x $7) = $1 to create the spread. If the stock price in 6 months is greater than $65 or less than $55, the total payoff is zero, and the investor incurs a net loss of $1. If the stock price is between $56 and $64, a profit is made. The maximum profit, $4, occurs when the stock price in 6 months is $6. Butterfly spreads can be created using put options. The investor buys two European puts, one with a low strike price and one with a high strike price, and sells two European puts with an intermediate strike price, as illustrated in Figure The butterfly spread in the example considered above would be created by buying one put with a strike price of $55, another with a strike price of $65, and selling two puts with a strike price of $6. The use of put options results in exactly the same spread as the use of call options. Put-call parity can be used to show that the initial investment is the same in both cases. A butterfly spread can be sold or shorted by following the reverse strategy. Options are sold with strike prices of KI and A" 3, and two options with the middle strike price K 2 are purchased. This strategy produces a modest profit if there is a significant movement in the stock price. Figure 11.7 Profit from butterfly spread using put options.

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